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Algebra Logika, 2007, Volume 46, Number 1, Pages 26–45 (Mi al7)  

This article is cited in 11 scientific papers (total in 11 papers)

Lattices of dominions of universal algebras

A. I. Budkin


Abstract: We fix a universal algebra $A$ and its subalgebra $H$. The dominion of $H$ in $A$ (in a class $\mathcal M$) is the set of all elements $a\in A$ such that any pair of homomorphisms $f,g:A\rightarrow M\in\mathcal M$ satisfies the following: if $f$ and $g$ coincide on $H$ then $f(a)=g(a)$. In association with every quasivariety, therefore, is a dominion of $H$ in $A$. Sufficient conditions are specified under which a set of dominions form a lattice. The lattice of dominions is explored for down-semidistributivity. We point out a class of algebras (including groups, rings) such that every quasivariety in this class contains an algebra whose lattice of dominions is anti-isomorphic to a lattice of subquasivarieties of that quasivariety.

Keywords: dominion, lattice of dominions, quasivariety.

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English version:
Algebra and Logic, 2007, 46:1, 16–27

Bibliographic databases:

UDC: 512.57
Received: 28.06.2006

Citation: A. I. Budkin, “Lattices of dominions of universal algebras”, Algebra Logika, 46:1 (2007), 26–45; Algebra and Logic, 46:1 (2007), 16–27

Citation in format AMSBIB
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\paper Lattices of dominions of universal algebras
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\pages 26--45
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. I. Budkin, “Dominions of universal algebras and projective properties”, Algebra and Logic, 47:5 (2008), 304–313  mathnet  crossref  mathscinet  zmath  isi
    2. A. I. Budkin, “Dominions in quasivarieties of metabelian groups”, Siberian Math. J., 51:3 (2010), 396–401  mathnet  crossref  mathscinet  zmath  isi
    3. Budkin A.I., “O dominione polnoi podgruppy metabelevoi gruppy”, Izv. Altaiskogo gos. un-ta, 2010, no. 1-2, 15–19  elib
    4. Budkin A.I., “O dominionakh konechnykh podgrupp”, Izvestiya Altaiskogo gosudarstvennogo universiteta, 2011, no. 1-2, 15–18  elib
    5. A. I. Budkin, “Dominions in Abelian subgroups of metabelian groups”, Algebra and Logic, 51:5 (2012), 404–414  mathnet  crossref  mathscinet  zmath  isi
    6. A. I. Budkin, “Absolute closedness of torsion-free Abelian groups in the class of metabelian groups”, Algebra and Logic, 53:1 (2014), 9–16  mathnet  crossref  mathscinet  isi
    7. A. I. Budkin, “On the closedness of a locally cyclic subgroup in a metabelian group”, Siberian Math. J., 55:6 (2014), 1009–1016  mathnet  crossref  mathscinet  isi
    8. S. A. Shakhova, “Absolutely Closed Groups in the Class of $2$-Step Nilpotent Torsion-Free Groups”, Math. Notes, 97:6 (2015), 946–950  mathnet  crossref  crossref  mathscinet  isi  elib
    9. A. I. Budkin, “Dominions in solvable groups”, Algebra and Logic, 54:5 (2015), 370–379  mathnet  crossref  crossref  mathscinet  isi
    10. A. I. Budkin, “On $2$-closedness of the rational numbers in quasivarieties of nilpotent groups”, Siberian Math. J., 58:6 (2017), 971–982  mathnet  crossref  crossref  isi  elib
    11. A. I. Budkin, “On dominions of the rationals in nilpotent groups”, Siberian Math. J., 59:4 (2018), 598–609  mathnet  crossref  crossref  isi  elib
  • Алгебра и логика Algebra and Logic
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